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A008931 Expansion of (2/(1+sqrt(1-36*x)))^(1/3). 1

%I

%S 1,3,45,936,22572,592515,16434495,473825700,14058408519,426438391743,

%T 13164565835421,412255067017248,13064028812911440,418149414542496168,

%U 13498863325944967656,439006511643775469856,14369623854340007790108,473027210589699351461700

%N Expansion of (2/(1+sqrt(1-36*x)))^(1/3).

%H G. C. Greubel, <a href="/A008931/b008931.txt">Table of n, a(n) for n = 0..640</a>

%F From _Vladimir Reshetnikov_, Oct 12 2016: (Start)

%F a(n) = 9^n*binomial(2*n + 1/3, n)/(6*n + 1).

%F D-finite with recurrence: n*(3*n+1)*a(n) = 6*(18*n^2-21*n+5)*a(n-1). (End)

%F a(n) ~ 2^(2*n-2/3)*3^(2*n-1)/(sqrt(Pi)*n^(3/2)). - _Ilya Gutkovskiy_, Oct 13 2016

%p seq(9^n*binomial(2*n +1/3, n)/(6*n+1), n=0..20); # _G. C. Greubel_, Sep 13 2019

%t CoefficientList[Series[Surd[2/(1+Sqrt[1-36x]),3],{x,0,20}],x] (* _Harvey P. Dale_, Aug 12 2016 *)

%t Table[9^n Binomial[2 n + 1/3, n]/(6 n + 1), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 12 2016 *)

%o (PARI) my(x='x+O('x^20)); Vec((2/(1+sqrt(1-36*x)))^(1/3)) \\ _G. C. Greubel_, Apr 11 2017

%o (MAGMA) I:=[1]; [n le 1 select I[n] else 6*(5-21*(n-1)+18*(n-1)^2)*Self(n-1)/((n-1)*(3*n-2)): n in [1..20]]; // _G. C. Greubel_, Sep 13 2019

%o (Sage) [9^n*binomial(2*n +1/3, n)/(6*n+1) for n in (0..20)] # _G. C. Greubel_, Sep 13 2019

%o (GAP) a:=[1];; for n in [2..20] do a[n]:=6*(5-21*(n-1)+18*(n-1)^2)*a[n-1]/((n-1)*(3*n-2)); od; a; # _G. C. Greubel_, Sep 13 2019

%K nonn

%O 0,2

%A _N. J. A. Sloane_

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Last modified July 15 14:12 EDT 2020. Contains 335772 sequences. (Running on oeis4.)